LMFDB - Newform orbit 1800.4.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$106.203438010$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 20 q^{7}+O(q^{10}) $

$ q - 20 q^{7} + 56 q^{11} + 86 q^{13} - 106 q^{17} + 4 q^{19} + 136 q^{23} + 206 q^{29} - 152 q^{31} - 282 q^{37} + 246 q^{41} - 412 q^{43} + 40 q^{47} + 57 q^{49} - 126 q^{53} - 56 q^{59} - 2 q^{61} + 388 q^{67} + 672 q^{71} - 1170 q^{73} - 1120 q^{77} + 408 q^{79} + 668 q^{83} - 66 q^{89} - 1720 q^{91} + 926 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−20.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$5$	$1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1800.4.a.f		1
3.b	odd	2	1		600.4.a.i		1
5.b	even	2	1		360.4.a.n		1
5.c	odd	4	2		1800.4.f.v		2
12.b	even	2	1		1200.4.a.p		1
15.d	odd	2	1		120.4.a.b	✓	1
15.e	even	4	2		600.4.f.a		2
20.d	odd	2	1		720.4.a.q		1
60.h	even	2	1		240.4.a.g		1
60.l	odd	4	2		1200.4.f.t		2
120.i	odd	2	1		960.4.a.bj		1
120.m	even	2	1		960.4.a.k		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.a.b	✓	1	15.d	odd	2	1
240.4.a.g		1	60.h	even	2	1
360.4.a.n		1	5.b	even	2	1
600.4.a.i		1	3.b	odd	2	1
600.4.f.a		2	15.e	even	4	2
720.4.a.q		1	20.d	odd	2	1
960.4.a.k		1	120.m	even	2	1
960.4.a.bj		1	120.i	odd	2	1
1200.4.a.p		1	12.b	even	2	1
1200.4.f.t		2	60.l	odd	4	2
1800.4.a.f		1	1.a	even	1	1	trivial
1800.4.f.v		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1800))$:

$ T_{7} + 20 $

$ T_{11} - 56 $

$ T_{17} + 106 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 20 $ T + 20
$11$	$ T - 56 $ T - 56
$13$	$ T - 86 $ T - 86
$17$	$ T + 106 $ T + 106
$19$	$ T - 4 $ T - 4
$23$	$ T - 136 $ T - 136
$29$	$ T - 206 $ T - 206
$31$	$ T + 152 $ T + 152
$37$	$ T + 282 $ T + 282
$41$	$ T - 246 $ T - 246
$43$	$ T + 412 $ T + 412
$47$	$ T - 40 $ T - 40
$53$	$ T + 126 $ T + 126
$59$	$ T + 56 $ T + 56
$61$	$ T + 2 $ T + 2
$67$	$ T - 388 $ T - 388
$71$	$ T - 672 $ T - 672
$73$	$ T + 1170 $ T + 1170
$79$	$ T - 408 $ T - 408
$83$	$ T - 668 $ T - 668
$89$	$ T + 66 $ T + 66
$97$	$ T - 926 $ T - 926
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Overview	Random
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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